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Published online by Cambridge University Press: 09 June 2025
Optimal protocols transform a given initial distribution into a given final one in finite time with a minimal amount of work or entropy production. We first analyze this optimization paradigmatically for a driven harmonic oscillator for which analytical results can be obtained. For a general Langevin dynamics, it is shown that the optimal protocol can be realized through a time-dependent potential with no need to use a nonconservative force. In contrast for discrete systems, nonconservative driving decreases the thermodynamic costs. For a broader perspective, we introduce concepts from information geometry which deals with the statistical manifold of distributions. The Fisher information provides a metric on this manifold from which the distance between two distributions as the minimal length connecting them can be derived. Speed limits yield relations between these quantities referring to an initial and a final distribution and the entropy production associated with the transformation of the former into the latter. For slow processes, cost along the optimal protocol or path is bounded by the distance between these distributions and the inverse of the allocated time.
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